Find the limit as t approaches infinity of T(t).

I have this problem and it seems pretty easy. I think I'm just overthinking it.

At time t = 0 minutes, the temperature of a cup of coffee is 180 degrees Fahrenheit. Left in a room whose temperature is 70 degrees Fahrenheit, the coffee cools so that its temperature function T(t), also measure in degrees Fahrenheit, satisfies the differential equation: dT/dt = -1/2(T) + 35.

Find . Explain what this means in the context of the problem.

What is it asking me to do when it says " ?" Is it asking me to set up an equation and solve (how would I do that?), or just explain what's happening? Also, can someone briefly explain what is happening? I'm stumped. Thanks for any help.

I
What is it asking me to do when it says "Find the limit as t approaches infinity of T(t)?" Is it asking me to set up an equation and solve (how would I do that?), or just explain what's happening? Also, can someone briefly explain what is happening? I'm stumped. Thanks for any help.

the limit as t approaches infinity of T(t)

It's asking you to solve the differential equation, obtain the function , and then take it's limit as right?

You can solve by separating the variables:

See if you can get to

then plug in the initial values to determine what is, then take exp of both sides, (or take exp of both side then figure out what c is) to get it to the form , then take the limit .

Dawg, you're getting the function name confused with the variable name . Keep them straight. So if I have and is an arbitrary constant from the integration, then if I add, subtract, multiply anything to it, it's still a constant. Then doing a little algegra, I get:

(still just an arbitrary constant c even though I took exponentials.

little more:

Now substitute the initial conditions to figure out that c=-110 to get: